learning mathematics: a case study: visual perception techniques in geometry
DESCRIPTION
This case study focuses on teaching preschool to kindergartner learners about mathematics through the use of visual perception techniques.TRANSCRIPT
Learning Mathematics 1
Running head: Learning Mathematics
Peaches M. Hubbard
Learning Mathematics: A Case Study: Visual Perception Techniques in Geometry
Jones International University, Online
Dr. Felicia Taylor
April 25, 2009
Learning Mathematics 2
Abstract
Kindergarten is where most students build a fundamental
knowledge of mathematics. Pre-school aged children are taught numbers
and shapes but usually through the process of recitation for memorization.
Kindergarten is where students improve upon their basic understanding of
what their numbers are, to how they can apply these numbers to create a
simple problem solving strategy. In kindergarten students learn about
numbers via pictures, manipulatives, class work and workbook content,
and through various artistic means, such as singing or game play. As
students get older they begin to see what learning style suits them, and
how they can incorporate that learning style into their class work or
studies. Kindergarten students may not be as advanced in their
understanding of what motivates them to learn, but through previous
experience and the knowledge of working with and having my own
children I have come to realize that children need visual stimuli to
promote learning. Therefore, the goal of this assignment is to discuss the
need and understanding of visual perception in mathematics; its uses, the
theories that support its use; and how to incorporate visual perception
techniques in mathematics curriculum.
Learning Mathematics 3
Table of Contents
Title Page 1
Abstract 2
Table of Contents 3
Introduction 4
Topic
Student Demographics 4
Student Population 5
Curriculum & Assessment Procedures 6
Mathematics Topics and Concepts
Concepts & Standards
Section III: Assessment Instruments 7 - 8
Case Study Background 9
Introduction to Assessment Instruments 10
Lesson Directives and Assessment Questions 11 - 14
Interview Questions 15
Visual Perception Techniques in Geometry 16
Section IV: Pre Assessment Results (Clean Copy of Pre-Assessment) 17 - 19
Written Assessment Rubric – Table I 20 - 26
Written Assessment Summary Rubric – Table II 27
Evaluation Summary 28 - 29
Instructional Plans and Activities
Lesson Plan I (P. 30 – 35)
Lesson Plan II (P. 36 – 42) 30 - 42
Instructional Plan Implementation 42
Ending Notes for Instructional Plan 43 - 44
Section VII - Post Assessment and Results
(Clean Copy of Post- Assessment) 45
Written Assessment Open-Ended Rubric – Table III 46 - 52
Summary of Open-Ended Response 53
Evaluation Summary 54 - 55
Student Work Samples (Post-Assessment) 56
Post Assessment Notes 57
Comparison Table (Post-Assessment Wrap-up) 58
Conclusion 60
JIU Consent Form 61
References 62 - 64
Learning Mathematics 4
Introduction
Course Project Topic
For the duration of this course project I will be working with a classroom of
kindergarten students that are placed in small cooperative groups. This is the third
quarter of school and the children were introduced to some first grade curriculum since
the second quarter. The topic that I have chosen for this course project is the use of
visual perception techniques in mathematics. In a study conducted in Columbus, Ohio,
one hundred and seventy-one second grade through sixth grade students were tested on
their visual perception. The assessment result concluded, “Poor visual perceptual ability
should be considered to be amongst the skills significantly related to poor mathematics
achievement, and that In fact, a significant relationship between visual perceptual skill
and mathematics abilities has been previously reported (Kulp, 1999; Solan, 1987). The
point of my project is to research and gather information regarding the use of visual
perception techniques in mathematics and to gain insight and knowledge into its use and
effectiveness in regards to kindergarten students.
School Demographics
For the duration of this course I will be working with an inner city charter school
located in Los Angeles, California, which is apart of the (LAUSD- Los Angeles Unified
School District). The charter school offers its students a quality education and upholds
the utmost standards of learning. The charter school consists of a pre-kindergarten and a
kindergarten program. As stated by the director of the school, “The goal of this
institution is to provide our students with a quality education; to introduce all
kindergarten students to first grade curriculum, by the second quarter of school, and to
Learning Mathematics 5
have our graduates move on to magnet programs.” (Mr. Richard Green- Charter School
Director)
The charter offers a commitment to both the students and the parent community.
Students are engaged through a well-rounded curriculum, creative arts programs (offered
through LAUSD), and the support of a caring and helpful staff. The schools founder was
is Sister Jennie Lechtenberg, who started off by offering an after school tutoring service
for low-income first and second graders. She them realized that a majority of the
students were missing the fundamental skills needed due to the language barriers faced
by their families. She then began to offer English classes, and so on. Today there are
two Puente learning centers. The state of the art facility also operates as a community
based non-profit and shares strong community ties by offering free classes to the
community, in addition to the charter school program. Via the facility classes are offered
to adults in computer technician training, ESL classes, Spanish classes, and
administrative classes. Classes are offered to youth as well, with after school clubs, math
academy, general tutoring, and a summer school for pre-kindergarten and kindergarten
students.
Student Population
The capacity for the charter school is sixty students. The pre-kindergarten and
kindergarten classes’ work in various rooms but as separate, and the class is comprised of
four classroom teachers. Students are moved through out three rooms, in which their
classroom activities take place. The students are offered math activities through class
lecture and exercises; a weekly home-work packet, and through computer-aided
programs (in a modern and up-to-date computer lab.) The students are placed in small
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groups of five to six students and each group is given a name. The groups are combined
of both boys and girls. For the duration of this class I will be working with the
kindergarten students, whom range in age from five to six. I will be present in the
classroom on Friday’s of each week. I will work with all of the kindergarten students,
and for the case study I will select one of the groups to work with one on one. There is a
strong relationship between parent and teachers, and students are assessed monthly on the
lesson, which they have worked on. Some students have transitioned to first grade level
curriculum, while others are receiving extra help in getting them prepared for the
transition to first grade. The learning style appears to be student based, providing inter-
active classroom with traditional and authentic lessons and assessment.
In-Class Math Instruction, Curriculum and Assessment Procedures
Students in the charter program are provided with a large group instruction,
individual group instruction, computer aided learning, and a weekly work packet. The
homework packet is given out every Friday and is due at the end of the following week.
Parent involvement is urged, in order to reinforce the skills that students learn in school.
At this time the students are working on single digit addition and subtraction, no
regrouping; counting; number writing practice; and number and number word
recognition. Students are not necessarily grouped by ability, but are pout in cooperative
groups and although the classroom teachers teach them, yet they are also encouraged to
assist each other, and to participate in the learning and teaching process.
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Course Project: Mathematics Topics and Concepts
Concepts and Standards
Concept one of my course project is to better understand a student’s factual
knowledge, when dealing in mathematics, versus their perceived knowledge of
mathematics. This deals with mathematics comprehension, and the need for the student
to not just memorize facts, but to use conceptual understanding and adaptive reasoning.
The activities that will be assigned for this concept include learning games and aides that
will help students bring fourth their innate skills for mathematics, while building on their
foundation of knowledge. First the students will be given assignments to complete
without instruction of the lesson, by using their innate mathematical ability. Next,
Students will be given comprehension questions regarding visual pictures, with
instruction, in which all five of the steps of mathematical proficiency will be utilized for
problem solving. Results will then be recorded and used accordingly. The goal of the
assessment is to see whether or not students have an innate ability to learn math; or with
proper instruction and tools can any students learn the process of mathematical problem
solving and comprehension? The above concept and activity ideas align with Standard
one of the Colorado State Department of Education’s standards for mathematics. This
standard states “Students develop number sense and use numbers and number
relationships in problem-solving situations and communicate the reasoning used in
solving these problems.” (CDE, 2005)
Concept two of my course project is to put into practice the techniques of visual
perception to determine its benefits in mathematics. “Visual perception has been used for
Learning Mathematics 8
centuries as an example in philosophical discussions about the nature of experience.
Traditional mathematical methods began to be applied to it in the second half of the
1800s,” (Stephen Wolfram, A New Kind of Science.) The focal point for this is algebra,
which is a related connection focal point for kindergarten mathematics. The focal point
connection states that for algebra “Children identify, duplicate, and extend simple
number patterns and sequential and growing patterns (e.g., patterns made with shapes) as
preparation for creating rules that describe relationships,” (National Council of Teachers
of Mathematics.) This would align with the connections standards for grades Pre-k
through second grade, which states, students at these grade levels should be able to
recognize and use connections among mathematical ideas; understand how mathematical
ideas interconnect and build on one another to produce a coherent whole; and recognize
and apply mathematics in contexts outside of mathematics (National Council of Teachers
of Mathematics.) This would be in accordance to standard two, which states: “Students
use algebraic methods to explore, model, and describe patterns and functions involving
numbers, shapes, data, and graphs in problem-solving situations and communicate the
reasoning,” and standard five, which states: “Students use a variety of tools and
techniques to measure, apply the results in problem-solving situations, and communicate
the reasoning used in solving these problems. (Colorado Department of Education) The
students will be assessed on a variety of activities in which they will be given visual
patterns, such as: completing the pattern using geometric figures, drawing the missing
parts (spatial awareness), visual discrimination, and (visual motor integration) copying
the picture. (edhelper.com) The above standards also coincide with the “Design
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Principles 3,5, and 6, for Fostering the Development of Whole-Number Sense.” (How
Students Learn: Mathematics in the Classroom, pp.292-302.)
In conclusion, visual perception is said to be a complex process, with about
eleven elements for pre-school through kindergarten students, these elements include: 1)
color perception and color constancy. 2) Shape perception and shape constancy. 3)
Spatial relations. 4) Visual analysis and synthesis. 5) Visual closure. 6) Visual
conceptualizing. 7) Visual discrimination. 8) Visual figure-ground distinction. 9) Visual
memory. 10) Visual pattern following. 11) Visual sequence. (Shirley’s Preschool
Activities, 2007.) I believe visual perception be an important factor in understanding
mathematics, especially with early primary grade level students. We can all gain-learned
knowledge, but we all perceive things differently, therefore I believe we must look
deeper into how a student learns, instead of how much a student can retain. It should not
be quantity versus substance in learning, the focus should on the goals and objectives,
how a students learns, and getting the student to learn to the best of their ability provided
of the best quality.
Section III: Assessment Instruments
Case Study Background
For the duration of this class I have aligned myself with a charter kindergarten class
located in Los Angeles, California. The program has a total of sixty students. I have
created a group of seven of the sixty students that I will be working with on a weekly
basis. The students are a differentiated group and come from various backgrounds; the
skill level of the students are comprised of (2) ESL learners, one students strives while
the other is a t a mid range learning level. The group is also comprised of a set of sibling
Learning Mathematics 10
twins a boy and a girl both on the low level range of learning. Both siblings have speech
impediments. The next sets of learners are both highly proficient students. Lastly, there
is a student whom I have added to the group; this student has behavioral concerns, but
has a willingness to learn. This student needs one-on-one attention and is at the mid to
low level range in his knowledge and skill base for mathematics.
Introduction to Assessment Instruments
For this assessment the students will be pulled out of class one by one and will not be
given instruction on the concepts of visual perception. Students will complete a packet
comprised of worksheets that focus on visual perception; the worksheets are authorized
for use by edhelper.com. Each student will be given a pre-assessment of ten questions, in
which they will communicate their responses both orally and written (fill-in-the-bubble.)
The goal of this assessment is to prove the benefits of visual perception skills in
mathematics to create a better understanding of learning. [“Learning, for visual-spatial
learners, takes place all at once, with large chunks of information grasped in intuitive
leaps, rather than in the gradual accretion of isolated facts, small steps or habit patterns
gained through practice. For example, they can learn all of the multiplication facts as a
related set in a chart much easier and faster than memorizing each fact independently."
(Study Guides and Strategies: The Visual Learner.)] The students will be given a set of
visual perception activities, these activities correspond the Colorado state standards as
well as the mathematics curriculum focal points of the National Council of Teachers of
Mathematics, which include a focus on algebra, geometry, and data analysis for
kindergarten students.
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Lesson, Directives, and Choice of Assessment Questions
Name
_____________________________ Date ___________________
Match the picture on top with one of the four choices.
1.
Match the picture on top with one of the four choices.
3.
Copy the lines shown on the left to the blank grid on the right.
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9. 10.
Circle the picture that is exactly the same as the picture on top.
13.
Complete.
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18.
How many times is the number 6 in the above picture?
4 times 5 times 10 times 2 times
Complete.
23. Draw the letter M in box C1.
Draw the letter C in box A1.
Draw the letter U in box B2.
Draw the letter E in box A3.
A B C
1
2
24. Draw the letter P in box A2.
Draw the letter K in box A1.
A B C
1
2
3
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3
Complete the pattern.
31.
?
A shape is missing from the square. Pick the shape that completes the square.
34.
(EdHelper.com provides all illustrations.)
Interview Questions
During the post assessment interview I will reiterate to the students that they will not be
graded on this assessment. I will review the goals of the assessment and review the
Learning Mathematics 15
questions that the student missed, if any. It is my goal to provide the students with
perception skills and techniques, as well as training in communication, grade level
appropriate mathematical vocabulary, and a better understanding of why we are
discussing and completing work for visual perception techniques. The students will be
asked the following questions in regards to their lesson:
What did you like or dislike about the assignment?
How often do you look at things at home and try to match them up or figure out
which piece of an object fits into another?
Have you had an eye appointment in the past?
What is the process or problem-solving skills that you used to figure out numbers
13, 31, and 34?
Did you have a hard time with questions 23 and 24? If so, why?
Which question did you like the best, and why?
Which was the hardest question for you, and why?
Were question 9 and 10 hard or easy? Can you tell me how you figured out how
to draw each shape? (i.e., did you count the dots or did you just look at the lines
and draw?)
Corresponding Standards
Standard Standard Rationale
Two Students use algebraic methods to explore, model, and describe
patterns and functions involving numbers, shapes, data, and graphs
in problem-solving situations and communicate the reasoning used
in solving these problems.
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Three Students use data collection and analysis, statistics, and probability
in problem-solving situations and communicate the reasoning used
in solving these problems.
Four Students use data collection and analysis, statistics, and probability
in problem-solving situations and communicate the reasoning used
in solving these problems.
(Colorado Department of Education, 1995.)
The Benefits of Visual Perception Techniques in Geometry
Visual perception is generally though of as plainly, how we see things. There is
so much more to more to visual perception, and the benefits of using visual perception
techniques in mathematics can greatly assist in student learning achievement. The
website Visual Learning for Life provides a detailed definition and description of what
visual perception is. This article and website relate directly to and supports the teaching
of the topic for this case study, the benefits of teaching visual perception techniques in
mathematics learning. A great quote that relates to this case study is from Aristotle;
“The soul never thinks without a picture” (Aristotle, Greek 384-322 B.C.) Teaching
about visual perception, especially to primary students will help them form and develop a
better understanding of interpretation, analysis, comprehension, and problem-solving
skills and techniques.
Section IV: Pre-Assessment Results
Clean Copy of Pre-Assessment Questions
Name: _____________________________________________
Match the picture on top with one of the four choices.
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1.
2.
Circle the picture that is exactly the same as the picture on top.
3.
Complete.
5.
How many times is the number 1 in the above picture? 9 times 7 times 4 times 10 times
Learning Mathematics 18
Draw a line from start to finish. Do not cross any lines.
6.
7. Draw the letter H in box A2. Draw the letter Y in box B2.Draw the letter N in box A1. Draw the letter L in box B1.
A B
1 2
8. Draw a circle in box A1.Draw a triangle box B2.
1 2
A B
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9.
?
10.
Written Assessment Open-ended Response Rubrics and Table 1
Student Name: Rochelle Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they X
(All illustrations are provided by Edhelper.com)
Learning Mathematics 20
created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 17
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
½ - DIDN’T ID SCALE
Problem Total 19 1/2
Student Name: Harry Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 21
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
½ Not
applicable
Problem Total 18 1/2
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
½ - DIDN’T ID SCALE
NotApplicable
Not applicable
Problem Total 15 1/2
Student Name: Joshua Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 22
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
XNot
ApplicableNot
applicable
Problem Total 15
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Student Name: Jafar Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 23
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
XNot
ApplicableNot
applicable
Problem Total 7
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 7
Student Name: Jennifer Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 24
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 4
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Student Name: Mary Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 25
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 3
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Student Name: Marta Date: 3/25/2008Question # 7 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 26
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 9
Question # 9 Conceptual Understanding
How well did the student:No evidence
0Some evidence
2Solid evidence
4Exceptional
6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 30
Table 2Summary Data of Open-Ended Response Rubrics
Rubric ScoreStudent Name
(First name or pseudonym)Question #
4Question #
9Additional Comments
Rochelle 17 19 ½
Strong Math Student (ESL Learner)#7 good use of concepts and procedures – procedurally correct, but completed only 3 out of 4 steps for answer.
#9 demonstrated a strong understanding of the problem.
Harry 18 ½ 15 ½
Harry id said to be a student who needs extensive help in mathematics and receives low assignment and test scores in mathematics.
#7 demonstrated a strong understanding of the problem.
#9 demonstrated a strong understanding of the problem.
Learning Mathematics 27
Joshua 15 24
Admits not knowing when to add, subtract, multiply, or divide
#7 Shoe an understanding of the problem, but failed to choose the correct answer.
#9 showed both strong conceptual and procedural understanding.
Jafar 7 7
Is classified as a student who struggles in math, and although he showed some conceptual and procedural understanding of questions 7and9, both questions appeared to be a challenge.
Jennifer 4 24
The sibling of Jafar also said to face serious challenges in mathematics.
#7 showed a lack of both conceptual and procedural understanding of the problem.
#9 Showed strong understanding of the problem.
Mary 13 24
#7 demonstrated little understanding of the problem as a whole.
#9 Showed exceptional understanding of this problem.
Marta 9 30
#7 showed some understanding of the problem.
#9 showed strong knowledge of the problem.
Learning Mathematics 28
The Evaluation Summary
The pre-assessment was conducted in a conference room located on the school
campus. My assessment group consisted of seven kindergarten students, whose range in
mathematics knowledge varies (From students who face challenges in mathematics, to
mid and high range level students, and English learner language students.) The students
have not had experience with this assessment material in class, but showed an openness
and willingness to take the assessment. Many of the students felt that the assessment was
fun. The students were pulled out of class, interviewed, and had the test conducted
individually. The teachers are pleased to introduce visual perception techniques in their
curriculum and I believe there are several benefits of this subject matter inclusion into
their mathematics curriculum.
There was no time allotment for the tests; each student completed the assessment
at their own pace. All of the students were very cooperative; they were given the
assessment instructions and completed their tests accordingly. The following is
documentation of each student’s start and finish time of the assessment, as well as any
notes that are valid for this exercise.
Start Time Completion Time
Rochelle 8:03 AM 8:24 AMHarry 8:27 AM 8:41 AMJoshua 8:49 AM 9:03 AMJafar 9:06 AM 9:27 AMJennifer 9:29 AM 9:49 AMMary 9:53 AM 10:11 AMMarta 10:13 AM 10:25 AM
Learning Mathematics 29
For question number seven of the assessment most students showed an
understanding of the concept of the boxes (A1, A2, B1, and B2), yet many were not able
to use their procedural understanding to discern between why and where they were to
place the letters, For example:
There were some misunderstandings related to the above problem, the majority of the
mistakes stem from difficulty with letter and number recognition. Box A and B 1 & 2
differentiation to name a few. Question number nine deals with geometry, the students
had to pick the missing portion of a shape, from three choices. Many students excelled
on this question, and while others did not choose the correct answer they demonstrated a
conceptual thought process and procedural understanding.
Learning Mathematics 30
Instructional Plans and Activities
Lesson Plan Title: “Having Fun With Learning Shapes – Part One”
Name: Peaches M. HubbardGrade: KindergartenNumber of Students: SevenAmount of time planned for the activity: Activity One: 15 Minutes/ Activity Two: 30 Minutes
Math Concept: Geometrical Shapes and Visual Perception Techniques.
Standard’s Covered
(Standard 2): Students use algebraic methods to explore, model, and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning used in solving these problems. (Standard 4): Students use geometric concepts, properties, and relationships in problem-solving situations and communicate the reasoning used in solving these problems. And, (Standard 6): Students link concepts and procedures as they develop and use computational techniques, including estimation, mental arithmetic, paper-and-pencil, calculators, and computers, in problem-solving situations and communicate the reasoning used in solving these problems.
Goals: The primary goals of this lesson are to:
Provide students with a better understanding and appreciation of shapes,
and to get a better understanding of how these shapes are formed.
To have students use their knowledge of shapes to see the shapes that are
around them in their daily lives.
For students to use their passion for drawing and to incorporate that in
their ability to draw various shapes.
Objectives: Recognizing various geometrical shapes names. Using color-coded geometrical shapes to review and identify various
geometrical shapes. Using numbers to help students count the sides of the shapes.
Learning Mathematics 31
Pre-assessment Findings: There were a few pre-assessment findings that led me to
believe that incorporating this lesson would give the students a better understanding of
geometrical figures. The first indicator that the students needed to work on shapes
occurred when several students showed some difficulty when they were asked to
complete the opposite side of a drawing. The other indicator that students showed
difficulty with shapes occurred when students were asked to draw shapes in the
corresponding box; most student’s were not able to draw these shapes and did not
recognize the shapes by name.
Description of Activity: The case study, kindergarten focus group will complete two
activities that will coincide with this lesson, they are as follows:
Activity One: students will be given a two-paged worksheet. Page one of the
worksheet will be comprised of geometrical shapes, by which students are to
recognize, call-out. And point to various geometrical shapes using their pointers.
Activity Two: Students will be provided with a dice template, which they are to
color, cut out, and play a shape recognition game.
Supplies/Materials Needed: The supplies needed for this lesson include: seven pencil and
erasers; chalk, pointers, and printed worksheets (3 worksheets per student.)
Teaching Strategies
Introduction: The students will be given instruction on recognizing shapes by
their names and by how many sides they have. For a demonstration, pictures of shapes
will be used, as well as real life objects that have the shame shape.
Learning Mathematics 32
Guiding/Discussion Questions:
1.) What are the shapes that you see around you, in the classroom?
2.) Using general classroom items, what shapes is this object?
3.) How many lines does it take to draw a. …(Various shapes)?
Checking for Understanding: The following are measures used to check for
understanding:
Reviewing concepts in an oral (group) review.
Using formative assessment through the guidance and discussion
questions.
Sending home a mixed worksheet, with all of the corresponding
activities to the lesson, and re-assessing the student’s worksheets.
Incorporation of Learning Principles Discussed in this Course: this lesson plan
incorporates the following learning principles:
Learning Principle One (Engaging Prior Understanding): this is incorporated in
this lesson by using various classroom items and identifying various “real world”
objects shapes. This will build on the foundation of what students know of shapes
while introducing them to other objects and shapes. (How Students Learn, p.4)
Design Principle One: Exposing Children to Major Forms of Number
Representation. According to the textbook, this principle is represented in five
major ways: through objects, dot set patterns, segments on a scale or graph, and
segments or points on a dial. This design principle will be used when the students
create the dice (object), as well as practice their counting when they roll the dice.
Design Principle Three: Providing Visual and Spatial Analogs of Number
Learning Mathematics 33
Representations That Children Can Actively Explore in a Hands-On Fashion.
Providing the students with fun learning activities that aid them in learning and
recognizing shapes and numbers fulfill the objectives of this design principle.
(How Students Learn, pp.283-293)
Learning Environments and Design Characteristics. All of the perspectives of
learning: (learning, knowledge, assessment, and community centered learning)
will create the environment for this lesson. The students will draw their own
shapes based on what they know of the shape, which is encouraged by learner-
centered environments. The students will be asked to create various geometrical
shapes step-by-step, in an attempt to understand how to make a shape, which is
encourages by knowledge centered learning environments. Students who are
demonstrating difficulty will be encouraged to do board work, in which they
will be given hands on demonstration of making shapes, which is encouraged by
assessment centered learning environments. Lastly, the students will free to ask
questions, and assist other students that may be having some difficulty with the
activities. This will also create a team effort or partnership, which can boost
self-esteem and create positive peer interaction. Community centered learning
environments encourages this type of learning. (How Students Learn, pp. 13-
17.)
Incorporation of the Five Strands of Proficiency (How Students Learn, p. 218):
The following describes how the five strands of proficiency will be incorporated
into the student’s lesson and learning activities.
Learning Mathematics 34
o Conceptual Understanding: comprehension of how many sides a shape
has.
o Procedural Fluency: learning how to draw various lines, angles, and
geometrical shapes.
o Strategic Competence: Understanding how to put together the sides to
make the dice.
o Adaptive Reasoning: N/A
o Productive Disposition: relating “real world” items and objects to see
how shapes surround us in our daily lives, and why objects of certain
shapes are used in the capacity that they are.
Incorporation of Constructivists Teaching Behaviors. The course website offers
this enlightening explanation of the constructivist classroom “It is important to
help all students construct new mathematical understandings, and at the same
time develop sensitivities to the wonder and benefit of understanding learners
who are different from themselves and who use various perspectives to solve
problems. “(JIU, EDU518: Teaching for Mathematics Comprehension.) I would
incorporate this idea into my lesson and corresponding activities by focusing on
what the students know about shapes, reinforcing that knowledge, and showing
them the proper way to create geometrical shapes.
Attachments: The following is a sample of the activities that correspond with this lesson.
Learning Mathematics 35
“Students will use pointers to point to the shape that is called out.”
Touch Worksheet (This is a sample of activity one. (MathWizardWorksheets.com)
Shapes Dice Template. (This is a sample of activity two.
(MathWizardWorksheets.com)
Lesson Plan Title: “Having Fun With Learning Shapes - Part Two.”
Name: Peaches M. HubbardGrade: KindergartenNumber of Students: SevenAmount of time planned for the activity: Activity I & II: 30 Minutes/Activity III: 15 Minutes
Learning Mathematics 36
Math Concept: Geometrical Shapes and Visual Perception Techniques.
Standard Covered: (Standard 2): Students use algebraic methods to explore, model, and
describe patterns and functions involving numbers, shapes, data, and graphs in problem-
solving situations and communicate the reasoning used in solving these problems.
(Colorado Department of Education, 2005)
Goals: The primary goals of this lesson are to:
Provide students with a better understanding and appreciation of shapes,
and to get a better understanding of how these shapes are formed.
To have students use their knowledge of shapes to see the shapes that are
around them in their daily lives.
For students to use their passion for drawing and to incorporate that in
their ability to draw various shapes.
Objectives: After the completion of the lesson the student’s will be able to
demonstrate their knowledge of:
How to draw various geometrical shapes, by drawing straight and angled
lines to create various geometrical shapes.
Using the knowledge that they have gained about shapes to complete
patterns.
Using their knowledge of numbers to count the correct number of shapes.
Recognizing various geometrical shapes.
Pre-assessment Findings: There were a few pre-assessment findings that led me
to believe that incorporating this lesson would give the students a better understanding of
Learning Mathematics 37
geometrical figures. The first indicator that the students needed to work on shapes
occurred when several students showed some difficulty when they were asked to
complete the opposite side of a drawing. The other indicator that students showed
difficulty with shapes occurred when students were asked to draw shapes in the
corresponding box; most student’s were not able to draw these shapes and did not
recognize the shapes by name.
Description of Activity: The case study, kindergarten focus group will complete two
activities that will coincide with this lesson, they are as follows:
Activity One: Students will be given two stapled worksheets, page one will be
comprised of geometrical shapes, by which students are to recognize, count, and
find the pattern for various geometrical shapes. Page two consists of a matching
activity by which students have to match the shape, with a part of the shape
missing.
Activity Two: Students will be given a worksheet, which they are to complete the
missing part of the picture by drawing the missing portion of the picture.
Supplies/Materials Needed: The supplies needed for this lesson include: seven pencil and
erasers; chalk, and printed worksheets (3 worksheets per student.)
Teaching Strategies
Introduction: The students will be given instruction on recognizing shapes by
their names and by how many sides they have. For a demonstration, pictures of shapes
will be used, as well as real life objects that have the shame shape.
Learning Mathematics 38
Guiding/Discussion Questions:
4.) What are the shapes that you see around you, in the classroom?
5.) Using general classroom items, what shapes is this object?
6.) How many lines does it take to draw a. …(Various shapes)?
Checking for Understanding: The following are measures used to check for
understanding:
Reviewing concepts in an oral (group) review.
Using formative assessment through the guidance and discussion
questions.
Sending home a mixed worksheet, with all of the corresponding
activities to the lesson, and re-assessing the student’s worksheets.
Incorporation of Learning Principles Discussed in this Course:
Learning Principle Two: The Essential Role of Factual Knowledge and
Conceptual Frameworks in Understanding. (How Students Learn, p.6.) This
principle will be incorporated into this lesson and its activities by teaching the
students about shapes and how to draw shapes, by understanding this factual
information students will be able to use their conceptual knowledge by finding
patterns: and understanding what shape comes next, as well as shape recognition,
and counting the sides and incorporating problem-solving skills.
Design Principle One: Exposing Children to Major Forms of Number
Representation. According to the textbook, this principle is represented in five
major ways: through objects, dot set patterns, segments on a scale or graph, and
segments or points on a dial. (How Students Learn, pp.283-284) This aligns
Learning Mathematics 39
with the activities of this lesson because the students will be provided with dot
set patterns to create shapes and practice number sense and counting skills. The
benefits of this are discusses by an online learning website, which states that
Geometry and Spatial Sense - Children build on their knowledge of basic shapes
to identify more complex 2-D and 3-D shapes by drawing and sorting. They then
learn to reason spatially, read maps, visualize objects in space, and use
geometric modeling to solve problems. Eventually children will be able to use
coordinate geometry to specify locations, give directions and describe spatial
relationships. (Time4Learning.com)
Learning Environments and Design Characteristics. All of the perspectives of
learning: (learning, knowledge, assessment, and community centered learning)
will create the environment for this lesson. The students will draw their own
shapes based on what they know of the shape, which is encouraged by learner-
centered environments. The students will be asked to create various geometrical
shapes step-by-step, in an attempt to understand how to make a shape, which is
encourages by knowledge centered learning environments. Students who are
demonstrating difficulty will be encouraged to do board work, in which they
will be given hands on demonstration of making shapes, which is encouraged by
assessment centered learning environments. Lastly, the students will free to ask
questions, and assist other students that may be having some difficulty with the
activities. This will also create a team effort or partnership, which can boost
self-esteem and create positive peer interaction. Community centered learning
Learning Mathematics 40
environments encourages this type of learning. (How Students Learn, pp. 13-
17.)
Incorporation of the Five Strands of Proficiency (How Students Learn, p. 218):
The following describes how the five strands of proficiency will be incorporated
into the student’s lesson and learning activities.
o Conceptual Understanding: comprehension of how many sides a shape
has.
o Procedural Fluency: learning how to draw various lines, angles, and
geometrical shapes.
o Strategic Competence: N/A
o Adaptive Reasoning: N/A
o Productive Disposition: relating “real world” items and objects to see
how shapes surround us in our daily lives, and why objects of certain
shapes are used in the capacity that they are.
Incorporation of Constructivists Teaching Behaviors. The course website offers
this enlightening explanation of the constructivist classroom “It is important to
help all students construct new mathematical understandings, and at the same
time develop sensitivities to the wonder and benefit of understanding learners
who are different from themselves and who use various perspectives to solve
problems. “(JIU, EDU518: Teaching for Mathematics Comprehension.) I would
incorporate this idea into my lesson and corresponding activities by focusing on
Learning Mathematics 41
what the students know about shapes, reinforcing that knowledge, and showing
them the proper way to create geometrical shapes.
Attachments: The following is a sample of the activities that correspond with this lesson.
Activity One: will be comprised of two worksheets. On the first worksheet
students are to count the shapes, which will help them with number and shape
recognition. On the second worksheet student’s are to use pointer to point to the
correct shape that is called out to them, each student can also take a turn calling
out the shapes for their classmates.
Counting Shapes Match-up
Worksheets
This is a sample of activity one. (MathWizardWorksheets.com)
Activity Two: this will be the third page of the lesson packet. Students are to
complete the missing part of the picture by drawing the missing portion of the
picture.
Learning Mathematics 42
This is a sample of activity two. (Edhelper.com)
Instructional Plan Implementation
The first lesson took place on Tuesday, April 14, 2009. The students were sitting
in their collaborative groups and were called individually to a table located in the math
and reading lab classroom. There are seven students that I am working with for this
assessment process. The student’s are comprised of high and low level learners, ESL
learners, and two siblings that have some developmental delays. The first lesson started
at approximately 7:45 a.m. and ended at 8:30 .m. The second lesson took place in the
same classroom on April 15, 2009, beginning at 7:50 to 8:45 a.m. The materials needed
for both lessons included: seven pencils, an eraser, crayons, and the activity worksheets.
The environment in the classroom was quite, most of the students were excited, and a few
were apprehensive because they were unsure of the activities that were in store for them.
Each lesson is comprised of two worksheets. For each lesson, I reviewed the
mathematical concepts of shapes. As a group we reviewed the name of the shape and the
sides, and gave examples of what things around them are of the same shape. Secondly, I
read all of the directions, and I after the review I had the students complete their work
individually. Lesson one asked the students to name and identify the correct shapes by
calling the shape out. The students then colored the shapes on their worksheet. The
second page of lesson one asked the students to draw a line and match the corresponding
Learning Mathematics 43
shape. For lesson two the students were asked to identify the shapes by name, count the
shapes, and fill-in-the blanks by counting the shapes and writing the proper amount on
the line. For the second page for lesson two the students were asked to draw the missing
half of the picture.
The students did not show any signs of boredom, they were excited and eager to
go through each worksheet. Although two of the more advanced students were able to
catch on to the concepts of learning quickly, all of the students took their time in
completing the lesson activities. The students interacted well with both each other and
myself throughout the duration of the lesson. The students were excited to learn and
answer questions, and did not hesitate to ask me questions, which let me know that they
were comfortable with their learning environment, with me, and with the lesson as a
whole.
Ending Notes:
The students were eager for me to work with them. Most of the students showed
some distress when we began page two of lesson two, in which the students were asked
to draw the missing half of the picture. They just assumed that they did not know how to
complete the task. [At this moment I stepped in and asked all of the students to at least
try, that there pictures were not expected to be perfect, but just to do the best that they
can. I also showed the students how they could use the concepts that they learned for the
day and the prior day to aid them in completing their pictures. One example that I
pointed out was of a picture of a castle. I pointed out all of the shapes that were shown
on the half of the picture that was complete. I explained that if they look at the shaped
they could see how to place them to draw the missing portion. For example, the topside
Learning Mathematics 44
of the caste was a triangle, which was connected to the bottom halve which was a
rectangle.] These simple suggestions seemed to help the students tremendously, they
became motivated and their self-esteem appeared to be boosted.
I ended each session with letting the students know that they all did a great job,
and each student received several stamps for their participation and hard work.
End Results:
There were a few students who are having some difficulties in their studies that
demonstrated some difficulty, I worked with these students and they were able to
complete the activities. The two sibling students who have pre-determined
developmental difficulty, were somewhat quite when they did not understand an activity
right away, although they did demonstrate their understanding by the end of the lesson.
The other students were equally involved in all of the activities. At the end of the
activities I reviewed the lesson and the student’s demonstrated their knowledge of being
able to identify, match, and give oral feedback regarding the shapes that were discussed
for the lesson. At the end all of the students demonstrated their knowledge of the lesson,
and for that reason I would not revise the lesson. The students were engaged,
comfortable, and excited to complete the activities associated with lesson one and two.
Learning Mathematics 45
Post Assessment and Results
Section VII: Post-Assessment and Findings
Clean Copy of Post-Assessment Questions
Name: _____________________________________________
Touch Worksheet: (MathWizardWorksheets.com)
“Students will use pointers to point to the shape that is called out.”
Learning Mathematics 46
Counting Shapes: (MathWizardWorksheets.com)“Students will demonstrate their knowledge of shapes by identifying, counting, and coloring the shapes on the worksheet. On the actual assessment document the student are provided with lines in which they are to fill-in-the blanks and write the number of shapes there are, for each shape.
Match-up Worksheet: (MathWizardWorksheets.com)“Students are to match the shapes on the left side of the worksheet to the shapes on the right side of the worksheet. Also, the students are to demonstrate their knowledge of shapes by naming each shape orally, as a group.
Complete the Picture: (Edhelper.com)“Students are to complete the picture by drawing the missing portion.”
Written Assessment Open-ended Response Rubrics and Table 3
Student Name: Rochelle Date: 4/15 - 16/2009Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
Learning Mathematics 47
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 20
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 22
Student Name: Harry Date: 4/ 15-16 /2008Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2 Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve
X
Learning Mathematics 48
the problem?
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
1/2
Problem Total 4 1/2
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 0
Student Name: Joshua Date: 4/ 15-16 /2009Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2 Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve
X
Learning Mathematics 49
the problem?
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Student Name: Jafar Date: 4/ 15-16 /2009Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence Some evidence Solid evidence Exceptional
Learning Mathematics 50
0 1 2 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 6
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 8
Student Name: Jennifer Date: 4/ 15-16 /2009Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4 Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a
X
Learning Mathematics 51
correct strategy.
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 13
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 20
Student Name: Mary Date: 4/ 15-16 /2009Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Learning Mathematics 52
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Student Name: Marta Date: 3/25/2008Question # 2 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1Demonstrate an understanding of the problem. Did the student draw a picture or write an expression or equation that properly represents the problem?
X
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 24
Question # 3 – Page 4 Conceptual Understanding
How well did the student:No evidence 0
Some evidence 2
Solid evidence 4
Exceptional 6
1 Demonstrate an understanding of the problem. X
Learning Mathematics 53
Did the student draw a picture or write an expression or equation that properly represents the problem?
2
Determine the relevant information in the question to solve the problem. Did the student write, circle, or otherwise indicate he/she knew the relevant information needed to solve the problem?
X
3Devise appropriate solution strategies to solve the problem. Is what the student wrote a viable way to solve the problem?
X
How well did the student:Procedural Understanding
No evidence 0
Some evidence 1
Solid evidence 2
Exceptional 3
4
Correctly implement the solution strategy/strategies to solve problem. Did the students solve the problem/strategy they created correctly? As we are evaluating procedural understanding, we are concerned with the solution to the problem/strategy the student created regardless if it is a correct strategy.
X
5Determine the correct answer to the original question. If the student was able to come up with the correct answer, give them a point regardless of their method.
X
Problem Total 30
Summary Data of Open-Ended Response Rubrics
Rubric Score
Student Name(First name or pseudonym)
Question # 2(Page 4)
Question # 3(Page 4)
Additional Comments
Rochelle 20 22
Strong Math Student (ESL Learner)#2 good use of concepts and procedures – procedurally correct, completed all the steps for the answer.
#3 demonstrated a strong understanding of the problem, and using shapes to help create a picture.
Harry 12 14
Harry is said to be a student who needs extensive help in mathematics and receives low assignment and test scores in mathematics.
#2 demonstrated some understanding of the problem.
#3 demonstrated no understanding of the problem. Henry needs extra help with developing his motor skills.
Joshua 24 24
Joshua is a bright student but he has behavioral concerns that interfere with his schoolwork; Joshua needs one-on-one attention.
#2 demonstrated a strong understanding of the problem and completed it appropriately.
#3 showed both strong conceptual and procedural understanding.
Jafar 6 8
Is classified as a student who struggles in math, yet he showed both, conceptual and procedural understanding of questions 2 and 3. Jafar had developmental concerns and functions at a much lower level of learning than the other students, yet he strived in both questions, which deal with art and shapes.
Jennifer 13 20 The sibling of Jafar also said to face serious challenges in mathematics.
#2 showed a great understanding of both
Learning Mathematics 54
conceptual and procedural knowledge of the problem.
#3 Showed strong understanding of the problem.
Mary 24 24
Mary is a fast learner and very bright, but as Joshua has behavioral concerns that can sometimes get in the way of her learning.
#2 showed a great understanding of both conceptual and procedural knowledge of the problem.
#3 Showed strong understanding of the problem.
Marta 24 30
#2 showed some understanding of the problem.
#3 showed strong knowledge of the problem.
The Evaluation Summary
The first lesson took place on Tuesday, April 14, 2009. The students were
sitting in their collaborative groups and were called individually to a table located
in the math and reading lab classroom. There are seven students that I am
working with for this assessment process. The student’s are comprised of high
and low level learners, ESL learners, and two siblings that have some
developmental delays. The first lesson started at approximately 7:45 a.m. and
ended at 8:30 .m. The second lesson took place in the same classroom on April
15, 2009, beginning at 7:50 to 8:45 a.m. The materials needed for both lessons
included: seven pencils, an eraser, crayons, and the activity worksheets. The
environment in the classroom was quite, most of the students were excited, and a
few were apprehensive because they were unsure of the activities that were in
store for them. For each lesson, I reviewed the mathematical concepts of shapes.
As a group we reviewed the name of the shape and the sides, and gave examples
of what things around them are of the same shape. Secondly, I read all of the
Learning Mathematics 55
directions, and I after the review I had the students complete their work
individually. Lesson one asked the students to name and identify the correct
shapes by calling the shape out. The students then colored the shapes on their
worksheet. The second page of lesson one asked the students to draw a line and
match the corresponding shape. For lesson two the students were asked to
identify the shapes by name, count the shapes, and fill-in-the blanks by counting
the shapes and writing the proper amount on the line. For the second page for
lesson two the students were asked to draw the missing half of the picture.
The students did not show any signs of boredom, they were excited and eager to
go through each worksheet. Although two of the more advanced students were able to
catch on to the concepts of learning quickly, all of the students took their time in
completing the lesson activities. The students interacted well with both each other and
myself throughout the duration of the lesson. The students were excited to learn and
answer questions, and did not hesitate to ask me questions, which let me know that they
were comfortable with their learning environment, with me, and with the lesson as a
whole.
The allotment for the tests: The students worked in a group and the
students had approximately thirty to thirty-five minutes per lesson. All of
the students were very cooperative; they were given the assessment
instructions and completed their tests accordingly.
All of the students did a good job for the lesson activity, yet Harry appeared to
struggle more than any of the other student with assessment questions two and three,
page four. Harry is a boy with a big personality, he is very friendly and outgoing, yet he
Learning Mathematics 56
tends to disturb or distract the other students when they are trying to learn. Henry has
some behavioral concerns, he is also lacking in the development of his fine motor skills.
Jafar did far better on this assignment, especially with questions two and three, yet he
continues to struggle with his motor and fine motor skills, he also has a speech
impediment and has a hard time speaking in complete simple and complex sentences.
Student Work Samples
Rochell
Rochell
Rochell
Learning Mathematics 57
Post Assessment Notes
The students did not show any signs of boredom, they were excited and eager to
go through each worksheet. Although two of the more advanced students were able to
catch on to the concepts of learning quickly, all of the students took their time in
completing the lesson activities. The students interacted well with both each other and
myself throughout the duration of the lesson. The students were excited to learn and
answer questions, and did not hesitate to ask me questions, which let me know that they
were comfortable with their learning environment, with me, and with the lesson as a
whole.
The students were eager for me to work with them. Most of the students showed
some distress when we began page two of lesson two, in which the students were asked
to draw the missing half of the picture. They just assumed that they did not know how to
complete the task. [At this moment I stepped in and asked all of the students to at least
try, that there pictures were not expected to be perfect, but just to do the best that they
can. I also showed the students how they could use the concepts that they learned for the
Learning Mathematics 58
day and the prior day to aid them in completing their pictures. One example that I
pointed out was of a picture of a castle. I pointed out all of the shapes that were shown
on the half of the picture that was complete. I explained that if they look at the shaped
they could see how to place them to draw the missing portion. For example, the topside
of the caste was a triangle, which was connected to the bottom halve which was a
rectangle.] These simple suggestions seemed to help the students tremendously, they
became motivated and their self-esteem appeared to be boosted.
Comparison Table
(The following table shows the results for both the pre-assessment and post-assessment results.)
Student Names Pre-Assessment
Scores
Post-Assessment
Scores
Total
Rochelle 36 42 78
Joshua 39 48 87
Jafar 14 14 28
Jennifer 28 33 61
Harry 33 1/2 26 59 1/2
Mary 27 48 75
Marta 39 53 92
Learning Mathematics 59
Post-Assessment Wrap-up
All of the students tried their best to follow the instructions given to them
regarding the assessment. Mostly, the same students that struggled in the pre-assessment
were the same students who faced some challenges in the post-assessment, due to pre-
determined learning and/or behavioral conditions. Most of the students showed
significant improvement from the pre-assessment to the post-assessment. The students
were able to demonstrate their knowledge of visual perception and geometry skills by
identifying shapes, counting, drawing, and using number sense and reasoning skills. The
students were exposed to the two of the curriculum focal points provided by the (NCTM),
which include:
Geometry, which states that kindergarten students should be able to “interpret
the physical world with geometric ideas (e.g., shape, orientation, spatial
relations) and describe it with corresponding vocabulary. They identify, name,
and describe a variety of shapes, such as squares, triangles, circles, rectangles,
(regular) hexagons, and (isosceles) trapezoids presented in a variety of ways
(e.g., with different sizes or orientations), as well as such three-dimensional
shapes as spheres, cubes, and cylinders. They use basic shapes and spatial
reasoning to model objects in their environment and to construct more
complex shapes.” (NCTM, 2005.)
Numbers and Operations, which states that kindergarten students should be
able to “use numbers, including written numerals, to represent quantities and
to solve quantitative problems, such as counting objects in a set, creating a set
Learning Mathematics 60
with a given number of objects, comparing and ordering sets or numerals by
using both cardinal and ordinal meanings, and modeling simple joining and
separating situations with objects. They choose, combine, and apply effective
strategies for answering quantitative questions, including quickly recognizing
the number in a small set, counting and producing sets of given sizes,
counting the number in combined sets, and counting backward.” (NCTM,
2005.)
Conclusion
It is my opinion that visual perception and geometry go hand in hand. Although
students may be able to learn various facts regarding numbers; their perception skills are
needed when it comes to demonstrating what they have learned through illustrations and
drawings to represent number values, shapes, or the like. Some students perceived
knowledge does not match up with the factual knowledge and this disconnect can hinder
a student in a variety of subjects, especially mathematics. As an educator I find it
imperative to teach students visual perception techniques; these techniques will not only
benefit the students in geometry but they provide strengthening exercises and strategies
for one’s mind and memory.
This case study has shed a new light on how students (kindergarten students) learn. I
have found new techniques and acquired new skills to help students have a better
understanding of mathematics. Therefore, I would like to end with a humorous quote that
can be used to explain the use of visual perception in mathematics: “Sometimes it is
useful to know how large your zero is. ~Author Unknown
Learning Mathematics 61
STUDENT WORK CONSENT AND RELEASE FORM
I hereby grant permission to Jones International University, Ltd., its affiliates and designees (collectively, JIU) to duplicate and use the material indicated below in the future without compensation to or consent from me. I acknowledge the duplication of the material may be in audio, digital, tangible print, internet-based or other forms of duplication and may be distributed in their entirety, abridged, compiled with other’s works or otherwise used by JIU. I expressly grant permission to JIU to use and distribute the same, as described above, as they elect, including as a part of its education courses. I also confirm I am the original author of the material or have otherwise identified the copyright owner or author in the material. I expressly release JIU and each of its affiliates and designees from and against all claims, demands, and causes of action that I may now have or in the future will have arising from their duplication and use of the material. I understand that, except for the rights granted to JIU above, I shall retain all ownership and other rights associated with the material.
Material:[Brief description of student’s project]
Case study regarding the benefits of using visual perception techniques in geometry.
Course: EDU518
Term: April 2009
Please sign this form below, indicating whether you accept or decline to have your work added to the JIU Course Project Library. Acceptance is optional and is in no way a requirement of your course.
ACCEPT DECLINE
Learning Mathematics 62
Signature: Peaches M. Hubbard Signature
Printed Name: Peaches M. Hubbard Printed Name:
Date: April 25, 2009 Date:
Please attach this form to your Course Project.
Reference:
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http://www.wolframscience.com/reference/notes/1076b
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Learning Mathematics 63
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Learning Mathematics 64
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Sample Instructional Implementation Plan. EDU518: Teaching for Comprehension. Dr. Felicia Taylor. Copyright © 2009 Jones International University®, Ltd. Centennial, C.O.
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Quote Garden (1998-2009.) Quotations About Mathematics. Last modified 2007 Oct 19 Fri 22:33 PDT. www.quotegarden.com/math.html